Pattern formation in three-dimensional reaction-diffusion systems

Existing group theoretic analysis of pattern formation in three dimensions [T.K. Callahan, E. Knobloch, Symmetry-breaking bifurcations on cubic lattices, Nonlinearity 10 (1997) 1179-1216] is used to make specific predictions about the formation of three-dimensional patterns in two models of the Turing instability. the Brusselator model and the Lengyel-Epstein model. Spatially periodic patterns having the periodicity of the simple cubic (SC), face-centered cubic (FCC) or body-centered cubic (BCC) lattices are considered. An efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamella, SC, FCC, double-diamond, hexagonal prism. BCC and BCCI states. Both models possess a special wavenumber k(*) at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams. This property is generic for two-species chemical reaction-diffusion models with a single activator and inhibitor (C) 1999 Elsevier Science B.V. All rights reserved.